Complete Minimal Varieties in Hyperbolic Space

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Given an oriented, C^ Riemannian manifold N, recall that the space of p- currents on N is (ß^*, the space of continuous linear functional on the space of p-forms, endowed with the weak topology If M^ is an oriented p-dimen- sional submanifold of N with compact closure and finite p-dimensional ume, then M defines a p-current [M] by

[ M ] ( a ) = ja, аейЦМ)

M

More generally, a rectifiable p-current is a convergent sum of currents ZjCMJ, where {M^}^ is a collection of mutually disjoint oriented p-rectifiable sets There is a natural mass norm defined on the space ^p{N) of rectifiable p- currents, given by

M ( r ) - sup { ^ ( w ) M(vi)^l}, S^edpiN), (11)

where M(w) = sup ||w^||, ||и^|| =sup{vi/^((^) ^ a simple unit p-vector} For

xeN

9^e , ß^ { N\ the total variation measure of ^ is defined as

\\9'\\ { A ) =M { 9'L . A )

where ^L^ denotes the restriction of ^ to Л (5^LЛ)(ш)= 5^(7^ ш), for /^ the characteristic function of Л The support of 5^ = I'[MJ is by definition

supp 9'=\j Mj Recall there is a natural boundary operator on (0^)* given by

{ dè^ ) { w ) =9' { dw )

One defines the space of integial p-currents ^p{N) on N to be the set of currents 6^Edp{N) such that дб^е^^^ ДАТ) It is clear that {J^^{NXd} is a chain complex We now state two of the major theorems of geometric measure theory

Theorem A [FF] For N a smooth Riemannian manifold, there is a natwal isomorphism

to singulai homology

Theorem В (Compactness Theorem) [FF] // KczN is a compact set and celR"^,

then the set

{ S^eJJN ) supp^czX, М(9')+М{д9')йс}

is compact in the weak topology

One easily sees that the mass function is lower semi-contmuous in the weak topology, thus one has a natural solution to the Plateau problem m the space of integral currents For instance, if ß^"^ is a (p-l)-dimensional submanifold (or integral (p —l)-current) such that B^~^ =c^9l for some 6^EJp{NX then there IS an integral p-current ^o with д9'^ = В and M{%)UM{'S^X V^ such that 09" = B